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About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as SIMULATION OF SHIP MANEUVERS USING RECURSIVE NEURAL NETWORKS 223 Simulation of Ship Maneuvers Using Recursive Neural Networks D.Hess, W.Faller (Naval Surface Warfare Center, Carderock Division, USA) ABSTRACT An improved Recursive Neural Network (RNN) maneuvering simulation tool for surface ships is described. Inputs to the simulation, cast in the form of forces and moments, are redefined and extended in a manner that more accurately captures the physics of ship motion; the new model is used to extend initial efforts toward RNN surface ship simulations. These extensions include improved formulations of propeller thrust, lift from deflected rudders, and the explicit inclusion of roll and pitch righting moments. Two maneuvers are simulated: tactical circles and horizontal overshoots. Simulation errors for the circles averaged over all maneuvers for such variables as speed, trajectory components and heading were 5% or less. The horizontal overshoot simulation errors were also 5% or less for the same variables with the exception of the transverse trajectory component. The explanation for the latter deficiency is believed to be the result of the exclusion of wind forces acting on the vehicle, which will be the subject of later work. INTRODUCTION The Neural Network Development Laboratory was established at the Naval Surface Warfare Center (NSWC) in 1995 with the directive to apply neural network technology as a predictive tool to problems of interest to the Navy. That such a tool might be successful was anticipated as a result of work using RNNs to predict and control three-dimensional unsteady separated flow fields (Faller, et al., 1995a) and dynamic reattachment (Faller, et al., 1995b). The subsequent development of an RNN-based simulation tool for submarine maneuvering was documented in (Faller, et al., 1997) and (Faller, et al., 1998a). RNN simulations have been created using data from both model and full-scale submarine maneuvers. In the latter case, incomplete data measured on the full-scale vehicle was augmented by using feedforward neural networks as virtual sensors to intelligently estimate the missing data (Hess, et al., 1999). The creation of simulations at both scales permitted the exploration of scaling differences between the two vehicles which is described in (Faller, et al., 1998b). More recent efforts have extended these techniques to the development of simulations of ship maneuvers. An initial formulation of the problem using an RNN model for use with ships is described in (Hess, et al., 1998). Speed and trajectory predictions for novel maneuvers segregated from the set used to train the network exhibited average errors of 5–8% in speed and trajectory components for tactical circle maneuvers. This level of error demonstrated the feasibility of the method, and showed that the gross features of the maneuvering behavior had been captured. Elsewhere, neural networks are in use or are planned in several applications. Notable among these is a neural network flight control system for a waverider subsonic vehicle (Saeks, et al., 1997). The neural control system will be used to control flight surfaces and to augment stability on the remotely piloted test vehicle. Another intelligent flight control system incorporating pre-trained and on-line trained neural networks (Totah, 1997) has been tested on a modified F-15 aircraft. The on-line network is designed to learn aircraft dynamics in real-time and detect unexpected changes in operation resulting from damage to the aircraft. Adaptive feedback provided to the controller might then allow reconfiguration for better performance under fault conditions. Neural networks have been used to assist a time-domain numerical model for prediction of pitch and heave motions of a trimaran frigate in regular head seas (Atlar, et al., 1997) and (Mesbahi and Atlar, 1998). Experiments conducted to complement the calculations were then used to train a neural network to predict pitch and heave motions for wave frequencies not contained within the training data. Also in the domain of ship research is the station-keeping (dynamic positioning) problem that has been considered by (Li and Gu, 1996). They devised a neural net controller to maintain horizontal position in the presence of wind and wave motions, and then tested it under simulated conditions within a computer. This paper extends the earlier ship simulation effort. Recursive neural networks have been trained to predict tactical circle and horizontal overshoot the authoritative version for attribution.

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About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as SIMULATION OF SHIP MANEUVERS USING RECURSIVE NEURAL NETWORKS 224 maneuvers for two surface ships. An RNN is a computational technique for developing time-dependent nonlinear equation systems that relate input control variables to output state variables. A recursive network is one that employs feedback; namely, the information stream issuing from the outputs is redirected to form additional inputs to the network. For this application, the RNNs are used to predict the time histories of maneuvering variables of full-scale vehicles conducting maneuvers in the open ocean. Full-scale data describing a series of maneuvers with varying rudder deflection angles and approach speeds have been acquired for each of two ships, and these data have been used to train and validate three neural networks, one for each ship and type of maneuver. Upon completion of training, data from maneuvers not included in the set of training maneuvers are input into the simulation, and predictions of the motion of the vehicle are obtained. The input data required for the model consists of time histories of the control variables: two propeller rotation speeds and two rudder deflection angles, along with the initial conditions of the vehicle at some prescribed starting location. As the simulation proceeds, these inputs are combined with past predicted values of the outputs to estimate the forces and moments that are acting on the vehicle. The resulting outputs are predictions of the time histories of the state variables: linear and angular velocity components which can then be used to recover the remaining hydrodynamic variables required to describe the motion of the vehicle. A schematic representation of the technique is shown in Fig. 1. Environmental data in the form of relative wind speed and direction are measured; future input forces and moments using these quantities are planned. Further details of the implementation of this model follow in subsequent sections, beginning with a description of the maneuvering data used for training and validation. Fig. 1 RNN surface ship simulation. DESCRIPTION OF DATA Data for training and validating the neural networks was acquired from two ships operating in the open ocean. Each ship is equipped with two propellers and two rudders. One ship is larger than the other such that differences in size, especially with respect to the rudders and propellers, make separate simulations of interest. Henceforward these ships will be referred to simply as Ship 1 and Ship 2. For each ship, the standard coordinate system attached to the center of gravity of the vehicle is assumed; namely, x is the longitudinal axis and is positive towards the bow, y is the transverse axis and is positive to starboard and the vertical axis z is positive downwards. Linear velocities in this coordinate system are indicated by u, v and w, and angular velocities by p, q and r. Accelerations are denoted similarly but with a dot above the letter to indicate a time derivative. Angles of roll, pitch and yaw are given by φ, θ and ψ, respectively. The speed of the vehicle is referred to by U. The vessel's controls are two propeller rotation speeds, n1 and n2, and two rudder deflection angles: δr1 and δ r2. Relative wind speed and direction will be indicated by VR and θR , whereas the true wind speed and direction will be denoted by V T and θT. These are the data that define the motion of the maneuvering vehicle, the controls that propel and direct the vessel and the details of the wind. They are summarized below in Table 1. The shading for each variable is added to indicate data that are directly measured, variables that may be derived from directly measured data and information not measured. For example, x and y, which refer to trajectory components relative to an inertial coordinate system placed at the starting location of a maneuver, are derived from latitude and longitude as measured by a global positioning satellite. Table 1 Required and measured data. the authoritative version for attribution. The neural networks have been trained to simulate two maneuvers: tactical circles and horizontal overshoots. The existing data are as follows. For Ship 1, 15 tactical circle maneuvers are available with rudder deflection angles which vary over a range of 10° to 35° and for a series of approach speeds from 5.1 m/s to 11.6 m/s (10 kn to 22.5 kn). Twenty-five tactical circle maneuvers are available for Ship 2 with rudder deflection angles from 10° to 35° and for approach speeds from 5.1 m/s to 15.4 m/s (10 kn to 30 kn). Because ships with multiple propellers often exhibit similar turning characteristics for both right and left turns, the bulk of the data are right turns with a small number of left turns. Ten horizontal overshoot ma00neuvers are available for Ship 2 with rudder

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About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as SIMULATION OF SHIP MANEUVERS USING RECURSIVE NEURAL NETWORKS 225 checking angles of 10° and 20° and for approach speeds from 3.4 m/s to 8.1 m/s (6.6 kn to 15.7 kn). A description of each of these maneuvers follows. Tactical circles are conducted in order to determine the inherent turning characteristics of the ship. Of particular interest are such quantities as advance, transfer, tactical and steady diameters, speed loss and steady speed in the turn. The typical procedure for these maneuvers is to first establish steady initial conditions for approach velocity and propeller rotation speeds and to maintain these conditions for 30 s. Then, an order to Commence Execution (COMEX) of the run is given. Data acquisition at a rate of 1 Hz begins and steady conditions are maintained for an additional 60 s. At this time an EXECUTE command is given and the rudders are deflected to the desired angle and maintained for the duration of the maneuver. These conditions are continued until the heading of the vehicle has changed by 540°; the maneuver is then terminated. An ideal tactical circle maneuver illustrating these terms is shown in Fig. 2. Note that Figs. 2 and 4 were reproduced from (Stenson and Hundley, 1989) and are used with permission. Fig. 2 Tactical circle maneuver. However, conditions in the open ocean are rarely ideal. Environmental factors such as moderate winds and unfavorable ocean currents can combine to produce the actual circle trajectory shown in Fig. 3. Knowledge of the time histories of wind speed and direction along with details of any prevailing ocean currents should allow one to formulate the appropriate force contributions that alter the trajectory of the vehicle. This is one of the areas in which future efforts will be directed. For the simulated maneuvers reported here, the problem was simplified by removing environmental effects. An automated procedure for removing the effects of drift was devised and applied to each of the maneuvers. This was possible because one assumes that the vehicle would indeed travel in a circle in the absence of environmental effects. Therefore, the trajectories were corrected by aligning overlapping portions of the circle and computing the unwanted drift velocity components which were then removed. An example of a corrected circle, rotated for convenience to place the steady state approach along the x-axis, is also shown in Fig. 3 Fig. 3 Typical actual and corrected tactical circle. Horizontal overshoot maneuvers are performed to characterize the handling response and rudder effectiveness of the vehicle, and such quantities as the heading overshoot angle, overshoot time, reach and period are used to establish this behavior. As with the circles, a 30 s period of steady initial conditions is followed by COMEX with a further 60 s elapsing before the EXECUTE order is given. After EXECUTE the rudders are deflected to a predetermined entrance angle and maintained in this position. The heading of the vehicle changes in response to the rudder deflection; when the heading has changed by a desired amount (typically equal to the entrance angle), the rudder is reversed and set to the rudder checking angle (usually equal to the entrance angle). Because the vessel does not respond instantly, the heading continues in the same direction for a period of time before slowing and then reversing. This overshoot angle measures the inherent ability of the ship to change direction. The procedure is typically repeated for 2.5 cycles. Unlike the circles, environmental effects cannot readily be removed from this data; however, the maneuvers are conducted such that the approach course is oriented the authoritative version for attribution. parallel (or anti-parallel) to the true wind direction. In this manner any influence of the wind on the ship's turning characteristics should be minimized for both left and right turns. Figure 4 shows superimposed plots of rudder deflection angle and heading during a typical overshoot maneuver and depicts useful terms. The neural network simulations were trained and validated using these maneuvers; a

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About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as SIMULATION OF SHIP MANEUVERS USING RECURSIVE NEURAL NETWORKS 226 description of the structure of the neural networks follows next. Fig. 4 Horizontal overshoot maneuver. RNN ARCHITECTURE The architecture of the neural network is illustrated schematically in Fig. 5. Fig. 5 Recursive neural network. The network consists of four layers: an input layer, two hidden layers and an output layer. Within each layer are nodes, which contain a nonlinear transfer function that operates on the inputs to the node and produces a smoothly varying output. The binary sigmoid function was used for this work; for input x ranging from −∞ to ∞, it produces the output y which varies from 0 to 1 and is defined by (1) Note that the nodes in the input layer simply serve as a means to couple the inputs to the network; no computations are performed within these nodes. The nodes in each layer are fully connected to those in the next layer by weighted links. As data travels along a link to a node in the next layer it is multiplied by the weight associated with that link. The weighted data on all links terminating at a given node is then summed and forms the input to the transfer function within that node. The output of the transfer function then travels along multiple links to all the nodes in the next layer, and so on. So, as shown in Fig. 5, an input vector at a given time step travels from left to right through the network where it is operated on many times before it finally produces an output vector on the output side of the network. Not shown in Fig. 5 is the fact that most nodes have a bias; this is implemented in the form of an extra weighted link to the node. The input to the bias link is the constant 1 which is multiplied by the weight associated with the link and then summed along with the other inputs to the node. For further details concerning the operation of neural networks, the reader is directed to (Haykin, 1994), and for recursive neural networks to (Faller, et al., 1997). A recursive neural network has feedback; the output vector is used as additional inputs to the network at the next time step. For the first time step, when no outputs are available, these inputs are filled with initial conditions. The time step at each iteration represents a step in dimensionless time, ∆t′ . Time is rendered dimensionless using the ship's length and its speed computed from the preceding iteration; thus, the dimensionless time step represents a fraction of the time required for the flow to travel the length of the hull. The neural network is stepped at a constant rate in dimensionless time through each maneuver. Thus, an input vector at the dimensionless time, t′, produces the output vector at t′+∆t′, where (2) the authoritative version for attribution. Because ship speed, U(t′ ), varies while length, L, is constant, the spacing between samples, ∆t, must vary in order that the dimensionless time step, ∆t′, remain constant at the chosen value of 0.09. The network described here has 56 inputs. Each hidden layer contains 54 nodes, and each of these nodes uses a bias. The output layer consists of 6 nodes, and does not use bias units. The network contains 118 computational nodes and a total of 6608 weights and biases. The input vector, described in detail below, consists of a series of forces and moments which act on the vehicle, and the network then predicts at each time step dimensionless forms of the six state variables: three linear velocity components u, v, and w, and three

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About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as SIMULATION OF SHIP MANEUVERS USING RECURSIVE NEURAL NETWORKS 227 angular velocity components p, q and r. Specifically, the outputs are defined as (3) These velocity predictions are then used to compute at each time step the remaining kinematic variables described in Table 1: trajectory components, Euler angles and accelerations. The 56 contributions that form the input vector are described as follows. Seven basic force and moment terms describe the influence of the control inputs and of time-dependent flow field effects: thrust from two propellers, Tstar and Tport, lift from two deflected rudders, Lstar and L port, two restoring moments resulting from disturbances in pitch and roll, Kr and Mr, and a Munk moment acting on the hull, NMunk. These terms are developed from knowledge of the controls: propeller rotation speeds and rudder deflection angles, geometry of the vehicle, and from output variables which are recursed and made available to the inputs. A detailed description of these seven inputs is reserved for the next section. Additional inputs are obtained by retaining past values of the seven basic inputs. This gives the network memory of the force and moment history acting on the vehicle and permits the network to learn of any delay that can occur between the application of the force or moment and the response of the vehicle. One past value from each of the two propeller thrust terms is retained to provide two additional inputs. For each of the remaining 5 basic inputs, 7 past values are retained as additional inputs. The number of past values to keep is chosen empirically and appears to be a function of the frequency response of the vehicle. In this case the network is given information about past events for a period of time required for the flow about the vehicle to travel a distance of 0.63L . Recursed outputs from the prior time step are used as six additional contributions to the input vector. Furthermore, the output vector from one previous time step is retained and made available as six additional inputs. Knowledge of the output velocities for two successive time steps permits the network to implicitly learn about the accelerations of the vehicle. A summary of the various contributions that make up the input vector is provided below in Table 2, and attention is next directed to a detailed explanation of the seven basic force and moment inputs. Table 2 Summary of network inputs. Input Description Inputs Tstar(t′), Tstar(t′−∆t′) 2 Tpart(t′), Tpart(t′−∆t′) 2 Lstar(t′), Lstar(t′−∆t′),…, Lstar(t′−7∆t′) 8 Lpart(t′), Lpart(t′−∆t′),…, Lpart(t′−7∆t′) 8 Kr(t′), Kr(t′−∆t′),… Kr(t′−7∆t′) 8 Mr(t′), Mr(t′−∆t′),…, Mr(t′−∆t′) 8 NMunk(t′), NMunk(t′−∆t′),…, NMunk(t′−∆t′) 8 u′(t′), v′(t′), w′(t′), p′(t′), q′(t′), r′(t′) 6 u′(t′−∆t′), v′(t′−∆t′), w′(t′−∆t′), 6 p(t′−∆t′), q′(t′−∆t′), r′(t′−∆t′) Total 56 FORCE AND MOMENT INPUTS Neural networks have an amazing ability to identify and track nonlinear behavior linking a set of inputs to a set of outputs. This innate ability can be further augmented, however, by carefully constructing physically motivated input and output variables that form a well-posed problem. For this task, inputs to the neural network were cast in the form of forces and moments acting on the vehicle: thrust from the propellers, lift from deflected rudders, restoring moments resulting from disturbances in pitch and roll and a Munk moment acting on the hull. These terms were fashioned from the basic control variables: propeller rotation speeds, n1 and n2; and rudder deflection angles: δr1 and δ r2. Also available for the definition of the input terms are output variables from the previous time step, which are recursed and made accessible to the input side of the network. In this manner a true simulation is preserved as only the control histories and initial conditions of the vehicle are required to run the simulation. The following paragraphs describe the creation of each of the input terms. The first two inputs to the network are thrust from the starboard and port propellers, respectively. A dimensional analysis (Lewis, 1988) relates propeller thrust coefficient, CT, to Froude number, Fr, advance ratio, J, cavitation number, σ, and Reynolds number, Re as follows: the authoritative version for attribution.

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About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as SIMULATION OF SHIP MANEUVERS USING RECURSIVE NEURAL NETWORKS 228 (4) where is the advance speed of the propeller, D is the propeller diameter and pv is the vapor pressure of the fluid. An approximation to this relationship may be found by considering only the primary dependence of the thrust coefficient upon the advance ratio. Expressing the thrust coefficient in the more convenient form, KT=T/ρ n2D4, a linear dependence upon J forms a useful approximation (Fossen, 1994), and may be stated as (5) Thrust may then be determined from (6) and the approximation becomes (7) The advance speed of the propeller is related to ship speed by means of the wake fraction, wf, and is written as UA=(1−wf) U, where wf ≈ 0.1 to 0.4. Furthermore, the presence of a hull-propeller interaction increases the resistance of the hull, or conversely, decreases thrust; therefore, T should be replaced by T(1−t), where t ≈0.05 to 0.2. Using these expressions, the thrust approximation may be written as (8) where the prop subscript is used to differentiate this basic propeller thrust from a reduction term to be discussed in a later paragraph. The propeller rotation speeds from the current time step and ship speed from the previous time step are used in Eq. 8 to generate the thrust approximation. Actual values for ρ and D were used and reasonable estimates of w f=0.2 and t=0.1 were made. The coefficients c1 and c2 were estimated for Ship 1. For Ship 2, thrust measurements were available for each of the maneuvers. For this case KT and J (computed using ship speed) were calculated for each point of the steady state portion preceding execution of the maneuver and then averaged. Plotting a point for each of the circle and overshoot maneuvers for Ship 2 revealed the trends shown in Fig. 6. Fig. 6 Measured KT vs. J for Ship 2. The coefficients c1 and c2 for each propeller for Ship 2 were taken from the fits to the data. These values and those estimated for Ship 1 are given in Table 3. Table 3 Coefficients c1 and c2. c1 -Star c2-Star c1 -Port c2 -Port −0.400 −0.400 Ship 1 0.600 0.600 −0.472 −0.460 Ship 2 0.553 0.533 Because the rudders are located aft of the propellers for these vehicles, a deflected rudder will interact with the propeller slipstream and reduce the overall thrust imparted to the vehicle (Söding, 1998). The drag is given by the authoritative version for attribution. (9) At a given time step, Eq. 8 is calculated which in turn allows a value for C T to be computed. Then, the reduction described by Eq. 9 may be determined, and the resultant thrust generated by the propellers may be computed from (10) This final expression was used to approximate the thrust imparted to the vehicle from the starboard and port propellers and represents the first two inputs to the neural network. The reader should note that although an attempt to produce reasonable coefficients for the thrust expressions was made, any errors in the coefficients will be accounted for by the network during training. The next two inputs to the network are lift forces generated by the starboard and port rudders,

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SIMULATION OF SHIP MANEUVERS USING RECURSIVE NEURAL NETWORKS 229 etting files. Pagebreak are true to the original; line lication as respectively. The lift force is approximated as the superposition of two contributions: lift on the rudder in the absence of a propeller, and an additional lift input resulting from deflection of the propeller slipstream (Söding, 1998). To describe the first contribution, a lift coefficient is required and is defined as Söding then approximates the lift hic errors may have been acident lly inserted. Pleas use the pr nt version of this pub on a rudder as (11) where α is an angle of attack. This formula uses the exact lift gradient for a thin foil in two-dimensional potential flow s given by i (12) e and then decreases it by a reduction factor which is a function of rudder aspect ratio that attempts to correct for deviations from two-dimensionality. It is defined as riginal paperbook, not from the original types (13) a The rudder aspect ratio, , may be found from =h2/Arud, where h is rudder height and Arud is rudder area. To compute the lift from Eq. 11, one requires the local angle of attack at the rudder. This quantity was calculated from c (14) When a transverse velocity component, v, or an angular velocity of yaw, r, is present, the simple result that angle of attack equals rudder angle must be corrected. The quantity Lr is the axial distance from the center of gravity to the rudder pivot. The denominator represents the longitudinal speed in the wake of the vehicle multiplied by an amplification factor due to the propeller. Söding estimated the additional lift on a rudder from an upstream propeller from momentum theory. The approximation is found from About this PDF file: This new digital repres ation of the or ginal work hasbeenrecompos from XML files creat d from the o (15) ome typograp Therefore, the total lift on the rudder is found as e (16) etting-spec formatting, however, cannotbe r tained, and s where Lrud is given by Eq 11 solved for lift, namely (17) e Summarizing, at a given time step, the angle of attack is computed from Eq. 14 requiring rudder angle, recursed values ed of transverse velocity, yaw angular velocity and ship speed, and a value for C T computed during the thrust calculations. The lift on the rudder is then computed from the sum of Eqs. 15 & 17, and these calculations are performed for the starboard and port rudders and form the next two inputs to the network. In addition to the propulsion and steering inputs two righting moment inputs are provided to account for disturbances in roll and pitch. A study of metacentric stability reveals that the righting arm in each case is proportional to the distance from the center of gravity to a point known as the transverse or longitudinal metacenter; this metacentric height is denoted or The product of the moment arm and the weight of the vehicle creates a couple which acts to restore the vehicle to its undisturbed orientation. These moments may be approximated by ific i (18) lengt s, word break headi g styles and ot er types where ∇ is volumetric displacement and φ and θ are angles of roll and pitch, respectively. The metacentric height is commonly decomposed into a difference between the distance from the center of buoyancy to the metacenter, or ent h and the distance from the center of buoyancy to the center of gravity, Furthermore, and may be approximated for small roll and pitch motions by IT/ ∇ and IL/ ∇, where IT and IL are moments of inertia of the wetted the authoritative version for attribution. portion of the vehicle about the transverse or longitudinal centerline, respectively. Upper bounds on these moments for , most ships satisfy IT<1/12B 3L and IL<1/12BL3, where B is the beam and L is the overall length of the vehicle. Replacing the fraction with a constant, the restoring moments may then be written as n s, (19) h

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SIMULATION OF SHIP MANEUVERS USING RECURSIVE NEURAL NETWORKS 230 etting files. Pagebreak are true to the original; line lication as This information, when available, can be explicitly provided, and the network will attempt to learn the unknown constants cT and cL naturally during training. Alternatively, the simpler expressions (20) hic errors may have been acident lly inserted. Pleas use the pr nt version of this pub may be used, again allowing the network to determine the unknown constants. These restoring moments were implemented in the latter form. Roll and pitch angles at the current time step were obtained by advancing previous roll and pitch angles using s i (21) e The derivatives and are obtained by using recursed angular velocities p, q and r and previously computed roll and riginal paperbook, not from the original types pitch angles from the equations (22) a The final basic input to the neural network is a moment which acts on the hull of the vehicle (Munk, 1920). c Consideration of a body moving through a perfect fluid and creating a potential flow reveals that fluid pressure acting on the hull will produce a net moment on the vehicle. A real fluid with viscosity and deviation from a potential flow introduces modifications but does not substantially change the resulting moment. The magnitude and direction of the moment depend on the square of the velocity of the vehicle and on the position of the vessel relative to the direction of motion. For a surface ship oriented with an angle of drift, β, relative to its direction of motion, the moment is given by (23) where VL and VT are fluid volumes having a mass equal to the added mass when the vehicle is oriented at β=0° and β=90°, respectively. For elongated bodies such as a ship, VT → ∇ and VL → 0. Therefore, the moment may reasonably be approximated by (24) About this PDF file: This new digital repres ation of the or ginal work hasbeenrecompos from XML files creat d from the o ome typograp The implementation of this moment as an input to the network requires ship speed and a recursed value of the transverse velocity, v, in order to compute β given by e (25) etting-spec formatting, however, cannotbe r tained, and s Any needed constant required to improve the approximation of Eq. 24 can be determined by the network during training. The seven basic inputs to the network have now been defined. They consist of the thrust from each propeller, the lift e from each deflected rudder, restoring moments to disturbances in roll and pitch and a Munk moment acting on the hull of the vessel. With the description of the architecture of the neural network now complete, attention is directed to the ed procedure by which the network was trained. TRAINING PROCEDURE As discussed in an earlier section, information presented to the inputs of a neural network is modified as it flows through the network by the presence of the weights and by the nonlinear outputs of each of the various nodes until it arrives at the output layer of the network. Thus, at each time step, an input vector produces a predicted output vector; this is then compared to the actual (target) output vector determined from the data. The difference between the target and predicted ific output vectors is a measure of the error of the prediction. The process by which the network is iteratively presented with an input vector in order to produce outputs that are then compared with a desired output vector is known as training. The i purpose of training is to gradually modify the weights between the nodes in order to reduce the error on subsequent iterations. In other words, the neural network learns how to reproduce the correct answers. When the error has been lengt s, word break headi g styles and ot er types minimized, training is halted, and the resultant collection of weights that have been established among the many connections in the network represent the knowledge stored in the trained neural net. Therefore, a training algorithm is ent required to determine the errors between the predicted outputs and the desired target values and to act on this information to h modify the weights until the error is reduced to a minimum. The most commonly used training algorithm, and the one the authoritative version for attribution. employed here, is called backpropagation which is a gradient descent algorithm. The collection of input and corresponding , target output vectors comprise a training set, and these data are required to prepare the network for further use. Data files containing time histories of tactical circle and horizontal overshoot maneuvers formed the training sets. n After the neural network has been successfully trained, the weights are no longer modified and remain fixed. At this point the network may be presented with an input vector similar to the input vectors in the s, h

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About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as SIMULATION OF SHIP MANEUVERS USING RECURSIVE NEURAL NETWORKS 231 training set (that is, drawn from the same parameter space), and it will then produce a predicted output vector. This ability to generalize, that is, to produce reasonable outputs for inputs not encountered in training is what allows neural networks to be used as simulation tools. To test the ability of the network to generalize, a subset of the available data files must be set aside and not used for training. These validation data files then demonstrate the predictive capabilities of the network. Three neural networks were trained in this manner to predict Ship 1 tactical circles, Ship 2 tactical circles and Ship 2 overshoots. In each case about 80% of the data files comprised the training set with 20% set aside as validation files. The networks were initially trained for 100,000 epochs, where an epoch is defined as the presentation of the time series for all inputs and outputs for all files in the training set. During this training process, training is paused every 10 epochs, and the network is tested for its ability to generalize. To carry this out, all of the files in the training set are combined with the validation files set aside earlier for this purpose, and the entire set is presented to the network. During this generalization phase, the weights are not modified; the data from the files simply go through the network to produce predicted outputs and these are then compared with the measured outputs. Use of the word output in this context implies any of the kinematic variables described in Table 1 which are computed at every time step. The errors are quantified by computing three error measures at each time step for each output. These errors are averaged over all of the outputs and then further averaged over all of the time steps in the file to produce measures of the generalization error for each file in this testing set. The errors are then further averaged over all of the files in the testing set (training files and validation files) to produce a measure of the generalization error over the entire set at this stage of training. The error measures are the absolute error and the dimensionless quantities: average angle measure and correlation coefficient. The average angle measure was developed by the Maneuvering Certification Board at NSWC and is similar to a correlation coefficient in that a value of one represents a perfect prediction and a zero value denotes a poor prediction. The equations defining these error measures may be found in (Hess, et al., 1999). After training has concluded, one examines the error measures as a function of the number of epochs that have elapsed. The curves reveal an optimum number of epochs at which training should have ceased and where minimum absolute errors and maximums in the dimensionless measures occur. Periodically during training, every 100 epochs, the weights are written to a data file. Neural network training is then restarted, at the closest epoch for which a weight set was saved, and continued to the desired optimum stopping epoch. Fig. 7 Evolution of generalization errors for Ship 2 circles. An example of the evolution of the generalization errors over the 100,000 epochs of the training phase for the network simulation of Ship 2 tactical circles may be found in Fig. 7. Of the 18 possible output variables described in Table 1 for which generalization errors may be computed, 5 were chosen as critical variables: u, x, y, φ and ψ. The generalization the authoritative version for attribution. performance of the network was judged on the basis of these critical variables for purposes of choosing the optimum stopping epoch. Because the relative magnitudes of each of these five variables are quite different, the averaged absolute error over the five variables will be a number that largely reflects errors in x and y, and this may be seen in the absolute error scale in the top graph of Fig. 7.

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About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as SIMULATION OF SHIP MANEUVERS USING RECURSIVE NEURAL NETWORKS 232 The stopping epoch for this network was selected as epoch 81,760. The network was restarted and then halted at this epoch. The absolute error was 45.1, the average angle measure was 0.953 and the correlation coefficient was 0.963. At this epoch the average angle measure reached its maximum, and the absolute error and the correlation coefficient were very close to their minimum and maximum, respectively. Figure 7 shows that after the first 20,000 epochs most of the points on each plot are clustered in a relatively thin band. This indicates that the solution is relatively smooth with small changes to the 6608 weights and biases at each epoch causing only minor fluctuations in errors. This was true for both Ship 1 and Ship 2 tactical circle simulations with the optimum solution typically appearing after 60,000 to 80,000 epochs of training. For the Ship 2 overshoot simulation, the optimum solution required a somewhat longer 100,000 to 150,000 epochs of training. The error plots were similar to Fig. 7 except that the points clustered in a thicker band. This would indicate that the evolution toward the optimum solution for the overshoots was more difficult with small changes to the weights causing larger fluctuations in solution errors. This topic will be revisited in a later section. Summarizing, three neural networks were trained to predict Ship 1 tactical circles, Ship 2 tactical circles and Ship 2 overshoots using the procedure described in this section. The results of these simulations proved to be quite encouraging and are detailed next. RESULTS Beginning with Ship 1, the network was trained using 12 tactical circles with three set aside for validation. Figure 8 depicts Ship 1 circle trajectories predicted by the trained network superimposed upon the actual trajectories followed by the vehicle. In each case the only information provided to the trained network was four time histories for port and starboard propeller rotation speeds and rudder deflection angles and the initial conditions of the vehicle. Comprising the top row of the figure are four of the 12 maneuvers used for training, and the bottom row contains all 3 validation runs. The four training runs that are shown represent a mixture of four different rudder angles and three different approach speeds. Similarly, the validation maneuvers contain three different rudder angles and two different approach speeds. Notice that two of the training circles that are shown are left turns with the other two right turns. Solid lines represent the predictions, and dashed lines are used for the actual path. In each case the steady state parts of the maneuvers after Comex and before Execute were reduced to a length determined by t′ =1, and only a portion of this straight path is shown. The predictions for the training circles are excellent. This is the first test that the network must pass. If a relationship between the force and moment inputs and the velocity outputs exists, the network must determine this connection. If the experimental data is poor, or the network is improperly formulated, then the network's performance on the training data will be correspondingly poor. Neither is the case here; the trained network has learned how Ship 1 performs a tactical circle maneuver. That this is true is evinced by the performance of the network on the validation circles. Recall that the validation runs were never used to modify the weights during training, and in this sense, have never before been seen by the network. The recursive neural network has been successfully able to generalize: to make predictions for maneuvers different from, but similar to, those represented in the training set. In two out of the three cases, the most difficult and nonlinear portion of the maneuver, the initial part of the turn as represented by the advance and transfer, has been captured precisely. Predictions for the steady turning diameter are excellent in all three cases. To quantify these statements, averaged errors for Ship 1 tactical circles have been tallied in Table 4 for the five critical variables: u, x, y, φ and ψ. The first number in each cell is an error averaged over all 15 maneuvers, whereas the second number is the error averaged over the 3 validation circles only. To give some percentage errors, the absolute errors were normalized by the following scales: average steady speed in the turn of 5.2 m/s (17 ft/s), average turning diameter of 651 m (2135 ft), average peak-to-peak roll variation of 2.8° and an average total heading variation of 529°. Table 4 Ship 1 tactical circle error measures averaged over all maneuvers/averaged over validation runs only. Var Abs. Err. Pct. Avg. Ang. Corr. Coef. u 0.24/0.32 m/s 4.6/6.2 0.975/0.968 0.978/0.961 x 22.6/47.9 m 3.5/7.3 0.985/0.969 0.996/0.986 y 13.7/26.2 m 2.1/4.0 0.975/0.960 0.997/0.993 φ 0.18/0.29° 6.3/10.2 0.931/0.870 0.939/0.898 ψ 3.2/4.6° 0.6/0.9 0.991/0.989 1.000/1.000 the authoritative version for attribution.

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About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as the authoritative version for attribution. 3050. Predictions: solid lines, Measured: dashed lines. and 3330. Predictions: solid lines, Measured: dashed lines. SIMULATION OF SHIP MANEUVERS USING RECURSIVE NEURAL NETWORKS Fig. 8 Ship 1 tactical circles. Top row: training files 3000, 3041, 3100, and 3070. Bottom row: validation files 3030, 3043, and Fig. 9 Ship 2 tactical circles. Top row: training files 3440, 3211, 3223, and 3460. Bottom row: validation files 3111, 3201, 3310 233

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About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as the authoritative version for attribution. Measured: dashed lines. SIMULATION OF SHIP MANEUVERS USING RECURSIVE NEURAL NETWORKS Fig. 10 Ship 2 tactical circle. Left column: training file 3223. Right column: validation file 3310. Predictions: solid lines, 234

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About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as SIMULATION OF SHIP MANEUVERS USING RECURSIVE NEURAL NETWORKS 235 Fig. 11 Ship 2 horizontal overshoots. Top row: training files 9002, 9040, 9030, and 9070. Bottom row: validation files 9060 and 9000. Predictions: solid lines, Measured: dashed lines. For Ship 2, the network was trained using 20 tactical circles with 5 set aside for validation, and Fig. 9 depicts the predicted and actual trajectories. The top row of the figure contains four of the 20 training maneuvers, 2 right turns and 2 left turns, and the bottom row contains 4 of the 5 validation runs. The four training runs that are shown represent a mixture of four different rudder angles and four different approach speeds, and the validation maneuvers consist of three different rudder angles and three different approach speeds. Even though Ship 2 is larger with bigger propellers and rudders the network again performs extremely well on the training runs and very good on the validation runs. Note that the fifth validation run is of similar quality, and was not shown due to space limitations. Table 5 Ship 2 tactical circle error measures averaged over all maneuvers/averaged over validation runs only. Var Abs. Err. Pct. Avg. Ang. Corr. Coef. u 0.12/0.18 m/s 1.5/2.3 0.989/0.984 0.987/0.968 x 37.9/55.7 m 3.5/5.2 0.982/0.978 0.993/0.984 y 29.3/40.0 m 2.7/3.7 0.967/0.948 0.995/0.990 φ 0.22/0.57° 9.7/24.9 0.840/0.523 0.845/0.462 ψ 4.4/7.2° 0.8/1.3 0.989/0.981 1.000/1.000 In three out of the four cases the advance and transfer, has been predicted extremely well. Predictions for the steady turning diameter are excellent in all four cases. The results for Ship 2 are quantified in Table 5. The first number in each cell is an error averaged over all 25 maneuvers, whereas the second number is the error averaged over the 5 validation circles only. The percentage errors were obtained by normalizing with: average steady speed in the turn of 7.9 m/s (26 ft/s), average turning diameter of 1072 m (3516 ft), average peak-to-peak roll variation of 2.3° and an average heading variation of 560°. Figure 10 displays Ship 2 speed, roll, pitch, heading and the direct neural network outputs: linear and angular velocity components. The left column of graphs gives the data for training file 3223 shown in Fig. 9, whereas the right column depicts graphs for validation circle 3310. One can see that the data is predicted almost perfectly for the case of the training file, and this is typical for all of the training files. For the validation file, agreement is again very good. In general, the predictions of v and w during the advance and transfer phase and especially φ are the most difficult quantities to predict. Ten horizontal overshoot maneuvers were also available for Ship 2. Eight were used to train the network with two set aside for validation purposes. the authoritative version for attribution.

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About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as the authoritative version for attribution. lines, Measured: dashed lines. SIMULATION OF SHIP MANEUVERS USING RECURSIVE NEURAL NETWORKS Fig. 12 Ship 2 horizontal overshoot. Left column: training file 9070. Right column: validation file 9060. Predictions: solid 236

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About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as SIMULATION OF SHIP MANEUVERS USING RECURSIVE NEURAL NETWORKS 237 The predicted and actual trajectories are shown in Fig. 11. The top row of the figure contains four of the eight training maneuvers, two beginning with right turns and two beginning with left turns, and the bottom row contains both validation runs. The four training runs that are shown represent a mixture of two different rudder checking angles and two different approach speeds, and the validation maneuvers consist of two different rudder checking angles and two different approach speeds. The horizontal overshoot is a more complex maneuver than the tactical circle and yet the network trained extremely well to the data. Small deviations from the actual path were typically 10 m or less with the largest deviations of about 20 m. This is also the case for the first of the two validation maneuvers. The prediction for the second validation run is exceptional until about 1000 m into the maneuver where a larger deviation of about 50 m occurs. The results for Ship 2 overshoots are listed in Table 6. The first number in each cell is an error averaged over all ten maneuvers, whereas the second number is the error averaged over the two validation runs only. The percentage errors were obtained by normalizing with: average steady speed late in the maneuver of 4.2 m/s (13.6 ft/s), average spatial period of 1424 m (4672 ft), average peak-to-peak distance of 153 m (500 ft ), average peak-to-peak roll variation of 1.2° and an average peak-to-peak heading variation of 41°. Table 6 Ship 2 horizontal overshoot error measures averaged over all maneuvers/averaged over validation runs only. Var Abs. Err. Pct. Avg. Ang. Corr. Coef. u 0.11/0.14 m/s 2.6/3.3 0.984/0.984 0.890/0.901 x 20.3/19.8 m 1.4/1.4 0.994/0.993 1.000/1.000 y 17.2/25.4 m 11.2/16.6 0.835/0.768 0.953/0.868 φ 0.09/0.24° 7.9/20.1 0.918/0.783 0.939/0.897 ψ 1.8/3.9° 4.5/9.5 0.893/0.981 1.000/0.944 Figure 12 illustrates Ship 2 speed, roll, pitch, heading and the neural network outputs: linear and angular velocity components. The left column of graphs gives the data for overshoot training file 9070 shown in Fig. 11, whereas the right column depicts graphs for validation run 9060. The agreement with the training data is excellent, and for the validation file, the agreement is very good indeed. Examination of Figs. 11 and 12 show that predictions of the important quantities defining a horizontal overshoot maneuver: reach, period, overshoot angle and overshoot time are predicted very well. The reader should note that errors in predictions from neural networks trained with experimental data can be no better than the precision error inherent in the data. To make some estimates of precision error, maneuvers with the same rudder deflection and approach speed were compared. For the tactical circles, the steady speed in the turn varied by 0.1–0.2 m/s (0.3–0.6 ft/s) or 1–4%, the turning diameter differed by 30 m (100 ft) or 3–5% and the peak-to-peak roll fluctuated by 0.2– 0.3° or 10%. For the horizontal overshoots, the estimated spatial period varied by as much as 150 m (500 ft) or 10%, the peak-to-peak distance fluctuated by 15 m (50 ft) or 10%, the peak-to-peak roll changed by 0.3–0.5° or 30–50% and the peak-to-peak heading differed by 1–3° or 2–7%. CONCLUSIONS Recursive neural networks trained on tactical circle maneuvers for two separate ships were able to predict speed, trajectory components and heading with errors averaged over all the data of 5% or less. When considering only the validation maneuvers, errors for these variables ranged from 1–7%. Errors in roll estimates were from 0.2–0.5° for an average roll variation of 2–3°. For the more complex overshoot maneuver, the errors were only a little higher. Speed, longitudinal trajectory component, x, and heading exhibited errors averaged over all the data of 5% or less, whereas consideration of just the validation maneuvers increased the errors to 10% or less. Roll errors were 0.1–0.3° for an average roll variation of 1.2°. The most difficult predictions for the overshoots were clearly the transverse trajectory component, y, an integral quantity resulting primarily from the transverse velocity component, v, and to a lesser extent, the heading. This is also the likely reason for the thicker band of generalization errors as the solution evolved during training. The effects of wind on the overshoot maneuvers will be felt strongly in these two variables as the maneuvers are always conducted parallel or anti- parallel to the wind direction. In the case of the circles, for which environmental effects could be removed, the networks performed well for these two variables. Clearly then, substantial improvement is likely to result from the addition of wind force and moment inputs to the neural network models, and future work will be directed to this end. Recursive neural networks have demonstrated an ability as a robust and accurate maneuvering simulation tool. With the further addition of environmental effects, the simulation could serve as a plant model within an adaptive control system. New control capabilities under investigation include the authoritative version for attribution.

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About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as SIMULATION OF SHIP MANEUVERS USING RECURSIVE NEURAL NETWORKS 238 trajectory following, maneuver pre-planning and improved station-keeping under adverse conditions. ACKNOWLEDGEMENTS The U.S. Office of Naval Research sponsors this work, and the program monitor is Dr. Teresa McMullen, Code 342. The support of Dr. Patrick Purtell, Code 333 is also gratefully acknowledged. The authors would also like to thank Mr. Richard Stenson and Mr. Donald Drazin of the Resistance and Powering Department of the Naval Surface Warfare Center for their assistance in acquiring the experimental data. REFERENCES Atlar, M., Mesbahi, E., Roskilly, A.P. and Gale, M. “Efficient Techniques in Time-Domain Motion Simulation Based on Artificial Neural Network,” International Symposium on Ship Motions and Manoeuvrability, RINA, London, U.K., Feb. 1998, pp.1–23. Faller, W.E., Schreck, S.J. and Luttges, M.W. “Real-Time Prediction and Control of Three-Dimensional Unsteady Separated Flow Fields Using Neural Networks”, Journal of Aircraft, Vol. 32, No. 6, 1995a. Faller, W.E., Schreck, S.J., Helin, H.E. and Luttges, M.W. “Real-Time Prediction of Three-Dimensional Dynamic Reattachment Using Neural Networks”, Journal of Aircraft, Vol. 32, No. 6, 1995b. Faller, W.E., Smith, W.E., and Huang, T.T. “Applied Dynamic System Modeling: Six Degree-Of-Freedom Simulation Of Forced Unsteady Maneuvers Using Recursive Neural Networks”, 35th AIAA Aerospace Sciences Meeting, Paper 97–0336, 1997, pp. 1–46. Faller, W.E., Hess, D.E., Smith, W.E. and Huang, T.T., “Full-Scale Submarine Maneuver Simulation,” 1st Symposium on Marine Applications of Computational Fluid Dynamics, U.S. Navy Hydrodynamic/ Hydroacoustic Technology Center, McLean, Va., May 1998a. Faller, W.E., Hess, D.E., Smith, W.E., and Huang, T.T. “Applications of Recursive Neural Network Technologies to Hydrodynamics”, Proceedings of the Twenty-Second Symposium on Naval Hydrodynamics, Washington, D.C., Vol. 3, August 1998b, pp. 1–15. Fossen, T.I. Guidance and Control of Ocean Vehicles, Wiley, New York, 1994, pp. 94–96, 246–248. Haykin, S. Neural Networks: A Comprehensive Foundation, Macmillan, New York, 1994. Hess, D.E., Faller, W.E., Smith, W.E., and Huang, T.T., “Simulation of Ship Tactical Circle Maneuvers Using Recursive Neural Networks,” Proceedings of the Workshop on Artificial Intelligence and Optimization for Marine Applications, Hamburg, Germany, September 1998, pp. 19–22. Hess, D.E., Faller, W.E., Smith, W.E., and Huang, T.T. “Neural Networks as Virtual Sensors”, 37th AIAA Aerospace Sciences Meeting, Paper 99–0259, 1999, pp. 1–10. Lewis, E.V., ed., Principles of Naval Architecture, Second Revision, Vol. 2, The Society of Naval Architects and Marine Engineers, Jersey City, 1988, pp. 127–153. Li, D. and Gu, M.X. “Dynamic Positioning of Ships Using a Planned Neural Network Controller,” Journal of Ship Research, Vol. 40, No. 2, June 1996, pp. 164–171. Mesbahi, E. and Atlar, M. “Applications of Artificial Neural Networks in Marine Design and Modeling,” Proceedings of the Workshop on Artificial Intelligence and Optimization for Marine Applications, Hamburg, Germany, September 1998, pp. 31–41. Munk, M.M. “The Aerodynamic Forces on Airship Hulls,” NACA TR-184, 1920, Reproduced in: Jones, R.T. “Classical Aerodynamic Theory,” NASA Reference Publication 1050, Dec. 1979, pp. 111–126. Saeks, R., Cox, C.J. and Pap, R.M. “LoFLYTE: A Neurocontrols Testbed”, 35th AIAA Aerospace Sciences Meeting, Paper 97–0085, 1997, pp. 1–6. Söding, H. “Limits of Potential Flow Theory in Rudder Flow Predictions”, Proceedings of the Twenty-Second Symposium on Naval Hydrodynamics, Washington, D.C., Vol. 2, August 1998, pp. 264–276. Stenson, R.J. and Hundley, L.L. “Performance and Special Trials on U.S. NAVY Surface Ships,” David Taylor Research Center Ship Hydromechanics Department Research and Development Report DTRC/SHD-1320–01, April 1989, pp. 60, 66. Totah, J.J. “Adaptive Flight Control and On-Line Learning” 35th AIAA Aerospace Sciences Meeting, Paper 97–3537, 1997, pp. 373–380. the authoritative version for attribution.

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About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as SIMULATION OF SHIP MANEUVERS USING RECURSIVE NEURAL NETWORKS 239 DISCUSSION J.Fein Naval Undersea Warfare Center, USA The neural net is a powerful interpolator of data. But by using estimated force and moment models, are you not introducing error and leading to a network that has the weaknesses of a coefficient model, rather than a neural net based purely on trials data with unspecified input. AUTHOR'S REPLY To answer this question, let us review a few details of the simulation technique. The inputs to the RNN model consist of time histories of the control variables and the initial conditions of the vehicle. To use the trained network, the user provides the required controls data, and a prediction of the motion of the vehicle is obtained. The control histories may come from experimental data, or alternatively, the control histories and initial conditions may be created by the user to describe a desired maneuver not carried out experimentally. The simplicity and flexibility of this arrangement makes the simulation easy to use. However, the resulting motion of the vehicle is not directly related to a rudder deflection angle or a propeller rotation speed but instead to the lift force produced by that deflected rudder or to the thrust produced by the propulsor. Therefore, an intermediate step, as shown in Fig. 1 in the paper, is required to transform easily specified input control variables into the forces and moments that direct the motion of the vehicle. Dr. Fein's question concerns the manner in which this transformation should be accomplished. This paper computed forces and moments from nonlinear expressions using empirical or fitted coefficients. The primary concern here is to define the shape of the nonlinear force and moment time histories as opposed to getting scaling factors exactly correct. Scaling the force and moment amplitudes or applying a DC offset will not affect the ability of the neural network to identify the relationship between force and moment inputs and motion outputs; instead, it will simply alter the magnitude of the weights that are determined during training. In fact, all of the training data must be scaled, using a linear transformation, to the domain and range of the activation functions employed within each node of the network (see Eq. 1 in the paper) before it can be used. For example, the complicated expressions for the righting moments defined in Eq. 19 can be replaced by the simpler expressions (R1) because the premultiplying coefficients simply serve to scale the amplitude of the sinusoids. More important here is to capture the fundamental sinusoidal behavior and then let the neural network determine during training the correct weight magnitudes to ensure appropriate scaling. On the other hand, coefficients such as c1 and c2 in Eq. 8 adjust the relative level of two contributions to Tprop and therefore help to define the shape of the resulting quantity. Correcting small deficiencies in these coefficients is likely to provide a second order correction to the predictions at best. The remarkable level of accuracy already obtained demonstrates this (generally in the 5–10% range, see Tables 4, 5 and 6). Furthermore, the accuracy that is ultimately achievable is constrained by the precision error inherent in the experimental data used to train the simulation. An alternative method for estimating resulting forces and moments from specified control variables is to employ CFD using potential or RANS codes. This may prove to be a superior method and has the added benefit that the geometry of the vehicle is explicitly defined during this process. Coupling CFD computations to the front end of a RNN simulation offers the potential for a geometry-to-motion simulation capability. The authors have already commenced efforts in this direction as part of another effort and work is proceeding. DISCUSSION T.Jiang Versuchsanstalt fur Binnenschiffbau e.V., Germany In your presentation, you apply partly very complete force formulations, for instance, for propeller and rudder forces, and partly very simple formula, for instance for Munk's moment. My question is what is important in modeling the hydrodynamics by using the neural network technique? Which details of the hydrodynamics are really required? AUTHOR'S REPLY In general, when constructing a neural network, one must include all inputs that have an influence on the outputs that one is trying to predict. Identifying the appropriate inputs can be a difficult task. For example, if the expression for the Munk moment can be significantly improved with the inclusion of free surface effects (as suggested in the discussion by V.Bertram below), then the output the authoritative version for attribution.

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About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as SIMULATION OF SHIP MANEUVERS USING RECURSIVE NEURAL NETWORKS 240 predictions are likely to improve. The level of improvement will be dictated by the degree to which the omitted input information affects the outputs. The fact that the work reported here achieved a level of accuracy generally in the 5–10% range (see Tables 4, 5 and 6) indicates that additional input information that may be lacking in the current simulation is likely to offer only second order improvements. On the other hand, for the horizontal overshoot maneuvers, the abnormally high error in the transverse trajectory component, y, is an indication that important input information is still missing. This is expected to improve with the addition of environmental factors (wind forces) as described in the paper. DISCUSSION E.Mesbahi University of Newcastle, United Kingdon Application of Artificial Neural Networks in dynamic modelling, simulation and control of highly nonlinear systems has been proved very successful for a wide range of industrial applications. Ranging from underwater remotely operated vehicles to the control and monitoring of satellite systems. Undoubtedly, marine technology and related industries, extending from preliminary ship design offices to the sophisticated ride control systems, all could benefit from ANN techniques as a versatile static and dynamic modelling platform. Authors are to be congratulated on a very concise and well presented paper. Their excellent work clearly shows their full appreciation of the physical rules governing ship motions. Recursive neural networks are used for dynamic modelling of ship manoeuvres. Comment 1: RNNs can provide non-physical models of dynamic systems; in other words, their application are similar to conventional system identification techniques such as Least Squares. An outstanding point in this paper is the combination of physical understanding of the ship motion parameters and RNN methodology. Authors force the RNN to learn the functional relationship between certain input parameters, which are calculated in “Component Force Modules”, and measured “6 DOF State Variables” (see Fig. 1). Once a valid model is obtained (after sufficient and successful training), it may be utilised for time-domain and consequently frequency-domain analysis of the ship motions. Question 1: Could we ignore the “Component Force Modules” box and try to model the ship motions by using “Environmental Wind and Waves” and “Control Signals” only? This is closer to a “Black-Box” modelling approach, where no physics is implied. Comment 2: The effect of the environmental factors such as wind sand unfavourable ocean currents are mentioned to cause drifts, hence disturbing the ship manoeuvring circles. This is partially corrected by an automated procedure. I hope this automated procedure looks after both tidal and wave and wind drifts. Question 2: Could we use these two signals, i.e. measured wave and wind, as additional inputs into RNN model and do not try to eliminate unavoidable drift in the manoeuvring circles? This may lead to modelling, or in better words, to finding the functional relationship between wave and wind signal and unconnected ship circles. Comment 3: Generalisation capability of RNNs are well described and presented Fig. 7 clearly shows that continuous training will improve the accuracy of the model for a particular ship type. Nevertheless, once the training is stopped, the RNN model only represents a specific ship type, ship1 or ship2 in this case. In other words, model is not a generic model capable of representing a wider range of ships. This is clearly a shortcoming of the RNN simulation when compared to conventional mathematical/physical modelling of systems. Question 3: Further research: What would happen if you mix the training epochs of ship1 and ship2 and train one single RNN? Obviously each set of data needs to be tagged separately by a numeric representing the ship number. But, is this generalisation possible at all? AUTHOR'S REPLY Question 1. The motion of the vehicle is not directly related to a rudder deflection angle or a propeller rotation speed but instead to the lift force produced by that deflected rudder or to the thrust produced by the propulsor. Therefore, the intermediate step transforming easily specified input control variables into the forces and moments that direct the motion of the vehicle is required However, the question is whether the neural network can learn this transformation or whether it must be explicitly defined. The authors did not try the suggested approach in this work. However, past the authoritative version for attribution.

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About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as SIMULATION OF SHIP MANEUVERS USING RECURSIVE NEURAL NETWORKS 241 experience with other problems in which both approaches were attempted has shown that output predictions can be improved by including known physical information. Particularly important is to define the type of input nonlinearity. For example, for the righting moments, which fundamentally have a sinusoidal response, the output prediction is likely to be better by explicitly defining inputs as sin φ and sin θ instead of just as φ and θ. Comment 2. The automated drift correction procedure does not require any knowledge about the cause of the drift. It simply corrects whatever drift is present. Question 2. The authors believe that the answer to this question is yes. Correcting for drift as opposed to directly predicting the motion of the vehicle executing unconnected circles has always been considered to be a temporary first step. The inclusion of environmental factors (currently underway) should make the drift correction procedure unnecessary. Comment 3. The fact that the current modeling scheme requires a separate RNN model for each ship type is disadvantageous. However, practically, the time required to create a different model for a different ship type is minimal if the experimental data is available. Therefore, the impact so far has been minor. As described above in the reply to J.Fein, the authors are attempting to inject geometry into the model by coupling CFD calculations on the front end. If this effort is successful, then a single model may be sufficient for a class of ships vis a vis a single ship. Question 3. This approach has not been tried and so the answer is not known. The likelihood is that the solution, if it converges, will represent neither ship. Minimal geometry information is included at present and is used primarily to render variables dimensionless. DISCUSSION V.Bertram Hamburg Ship Model Basin Germany I congratulate the authors on their progress made since the 22nd Symposium on Naval Hydrodynamics. The authors remind us with their interesting contribution that indeed ship hydrodynamics encompasses more than just CFD and progress is still possible (and thus research worthwhile) also in experimental fluid dynamics. The authors succeeded again in presenting their work in a clear and unpretentious way that serves as a good introduction to the neural network technology. I hope the paper will inspire more colleagues to study and incorporate neural networks in ship hydrodynamics. I would like to add a few references for neural network applications in ship hydrodynamics. Various applications including the trimaran ship motion prediction can be found in [2] which should be more widely available than the 1998 reference. [3] use neural networks to predict rudder forces on a new integrated propeller-rudder system. [4] employ neural networks to derive simple ship design estimates especially for tugs. [6] has developed a commercial program for the prediction of propeller induced pressure pulses accounting also for cavitating propellers. [1] use neural networks in controlling fins to reduce ship motions where the neural network is trained on new seaways. Apparently neural networks drift slowly into the maritime industry but are far from being widely used. I would like to invite the authors' comments on why we do not see more researcher employing neural networks and what could be done to disseminate the technology faster within the community. On several occasions the authors have introduced more knowledge about the physics than in the previous work presented at the 22nd Symposium in Washington. The pragmatic engineering approach obviously improves the overall modelling capabilities. Coefficients are often simply estimated by constants lying in a typical bandwith of ship data. The authors claim that “although an attempt to produce reasonable coefficients for thrust expressions was made, any errors in the coefficients will be accounted for by the network during training.” This statement is somewhat misleading. Neural nets can only yield proper predictions if all relevant input variables are supplied. This is acknowledged in the explanation of remaining errors due predominantly to the not yet modelled influence of wind, but should be stressed here again. In addition, neural nets map functions on intervals between 0 and 1. If the actual data lie only on a small subinterval, the “resolution” of the approximation will be worse than if the whole interval would be used. This should result in higher errors in the prediction of the trained network. It is therefore desirable to supply as much information as possible or reasonable to the model and reduce the “brute force” neural net system identification to the remaining coefficients. Wake fraction, thrust deduction coefficients and stability coefficients like may be estimated with more accuracy the authoritative version for attribution.

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About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as SIMULATION OF SHIP MANEUVERS USING RECURSIVE NEURAL NETWORKS 242 depending on ship type or main dimensions. [5] give various empirical formulae derived by conventional regression analysis. Using better estimates here may facilitate the remaining job for the neural nets. The Munk moment seems to be predicted quite simply neglecting the effect of the free surface. Even within potential theory, the ship will experience a trim moment and a vertical force due to the effect of the free surface. These forces will usually increase with speed and decrease with water depth. They may be important in shallow water even at low speeds. Such forces can be predicted reasonably well with free-surface potential flow codes requiring perhaps 30 minutes CPU time on last generation PCs. I would like to invite a comment on whether the authors deem an extension of their model in this direction as worthwhile or whether the expected improvement would not justify the added complexity in the approach. As a final comment: Have the authors considered to validate their neural network approach against ship model tests under controlled conditions in towing tanks? Model experiments should then be much better reproduced if indeed the influence of wind is the predominant factor accounting for errors in model. 1. Liut, D.A., Mook, D.T., VanLAndingham, H.F. and Nayfeh, A.H. “Roll reduction in ships by menas of active fins controled by a neural network”, Ship Technology Research 47, 2000, pp. 79–89. 2. Mesbahi, E. and Atlar, M. “Artificial Neural Networks: Applications in Marine Design and Modelling”, 1st International Conference on Computer Applications and Information Technology in the Marine Industries, COMPIT'2000, Potsdam, 2000, pp. 276–291. 3. Mesbahi, E. and Koushan, K. “Empirical Prediction Methods for Rudder Forces of a Novel Integrated Propeller-Rudder System”, OCEANS'98, IEEE/ OES Conference, Nice, 1998, pp. 532–537. 4. Mesbahi, E. and Bertram, V. “Empirical Design Formulae using Artificial Neural Nets”, 1st International Conference on Computer Applications and Information Technology in the Marine Industries, COMPIT'2000, Potsdam, 2000, pp. 292–301. 5. Schneekluth, H. and Bertram, V. “Ship Design for Efficiency and Economy”, Butterworth & Heinemann, Oxford, ISBN 0750641339, 1998. 6. Koushan, K. “Prediction of Propeller Induced Pressure Pulses Using Artifiical Neural Networks”, ”, 1s t International Conference on Computer Applications and Information Technology in the Marine Industries, COMPIT'2000, Potsdam, 2000, pp. 248–254. AUTHOR'S REPLY Comment 1. Neural networks are likely to become more widely used within the maritime community if they can be shown to be successful and if they can demonstrate that they present a viable alternative to existing techniques. A large fraction of extant neural network research reported in the literature has been concerned with academic questions regarding textbook problems as opposed to exploring applied problems. See the literature review by Faller, 1996. Increased application of neural networks to applied problems is likely to alleviate both concerns. Comment 2. The authors agree that the inclusion of relevant physical information to reduce the level of brute force system identification that must be performed by the neural network is desirable. However, efforts to better estimate those coefficients that merely scale the amplitude of the input are not required as described in the reply to J.Fein. An additional example of such a scaling coefficient is the thrust reduction coefficient as implemented in Eq. 8. Using better estimates for those coefficients that determine the shape of the input time histories, such as wake coefficient, may indeed improve the results somewhat. Comment 3. The use of CFD codes to augment or directly perform the calculation of force and moment inputs is an excellent suggestion that has been considered. Initial attempts to couple CFD calculations with the RNN simulation are underway as part of another project. Comment 4. The authors have considered using ship model tests to validate the neural network approach. We are attempting to identify appropriate experimental data that contains enough measured variables to be useful for this purpose. REFERENCE Faller, W.E. and Schreck, S.J. “Neural Networks: Applications and Opportunities in Aeronautics”, Progress in Aerospace Sciences, Vol. 32, No. 5, 1996. the authoritative version for attribution.